It is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration $C_{14}$. We find three configurations such that a Steiner triple system is affine if and only if it does not contain one of these configurations. Similarly, we characterise Hall triple systems using two forbidden configurations. Our characterisations have several interesting corollaries in the area of edge-colourings of graphs. A cubic graph G is S-edge-colourable for a Steiner triple system S if its edges can be coloured with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. Among others, we show that all cubic graphs are S-edge-colourable for every non-projective non-affine point-transitive Steiner triple system S.

We introduce the notion of strongly projective graph, and characterise
these graphs in terms of their neighbourhood poset. We describe certain
exponential graphs associated to complete graphs and odd cycles. We
extend and generalise a result of Greenwell and Lov\'asz \cite{GreLov}:
if a connected graph $G$ does not admit a homomorphism to $K$, where $K$
is an odd cycle or a complete graph on at least 3 vertices, then the
graph $G \times K^s$ admits, up to automorphisms of $K$, exactly $s$
homomorphisms to $K$.